The Greek Orthodox Church has a feast of the Dormition of Mary--her "falling asleep"--also celebrated on August 15. They believe her body was taken up to heaven. The main difference is that the Orthodox say that Mary definitely died, whereas Pius XII left it open when he defined the dogma of the Assumption, whether Mary died or not before being assumed into heaven.
This is from what appears to be the official website of the Greek Orthodox Church in America--"dormition" is the Latin equivalent of the Greek term for falling asleep.